Intermediate Friction Research Articles

Two novel approaches to the Kramers rate problem in the spatial diffusion regime

At present, there are two general theoretical approaches to calculating the rate of thermally activated escape of a Brownian particle over a barrier out of a metastable well in the spatial diffusion regime. A direct approach involves techniques entirely based on the underlying Fokker–Planck equation, such as the Kramers flux over population method, the mean first passage time formalism, and the eigenmode expansion. An alternative consists of replacing the original one-dimensional stochastic dynamics by an infinite dimensional Hamiltonian system. The rate is then calculated using reactive flux methods. Both approaches are rather efficient when treating bistable potentials with high parabolic barriers. However, complications arise if the barrier is not parabolic. In such a case, large deviations of theoretical predictions from exact numerical rates are observed in the intermediate friction region. The latter holds true even though the barrier is infinitely high, to say nothing of low barriers for which the problem of finite barrier height corrections remains effectively unresolved. Based on the expansion of the Fokker–Planck equation in reciprocal powers of the friction coefficient, two novel methods for calculating analytically the rate of escape over an arbitrarily shaped barrier are presented. These are a continued fraction expansion method and a self-similar renormalization technique developed recently for summation of divergent field-theoretical series, respectively. In this way, two different rate expressions are constructed that agree in the limiting case of high friction with the rate following from the corresponding Smoluchowski equation and reduce to the transition state theory rate at zero damping. Comparison with a known rate expression for a purely parabolic barrier and from numerical simulations for bistable potentials with cusped and smooth barriers of different heights show excellent agreement between the present theories and exact numerical results. As long as the escape dynamics is dominated by spatial diffusion across the barrier top, the maximal relative errors attained with the continued fraction method and the self-similar renormalization technique are less than 3% and 7%, respectively. This is in drastic contrast to known rate formulas derived by other means, whose relative errors are larger by factors and even by orders of magnitude.

Vibrational dephasing of an anharmonic solute strongly coupled to solvent

We present new results for an old model: an anharmonic solute linearly coupled to a harmonic bath. A comparison of numerical simulations of the classical mechanical absorption spectrum to the results of conventional perturbation theory and the instantaneous normal mode (INM) approximation shows that the INM performs poorly in the low-friction regime, but yields reasonable results in the high-friction regime, in which perturbation theory is inappropriate. An analytical theory for the line shape is formulated to describe the regime of intermediate friction, in which neither perturbation theory nor the INM approximation works well.

Conductivity of two-dimensional systems of interacting Brownian particles within the effective potential description

We investigate the dynamic properties of two-dimensional systems composed of N interacting Brownian particles subject to a periodic potential. By employing the Langevin dynamic simulations and assuming a repulsive Yukawa potential type, we have computed the effective potential V eff. Our results show that for the incommensurate concentration c=2/3, the barrier height of V eff increases in one direction and decreases in the other compared with the that for the non-interacting particles case. This behaviour explains quantitatively the aspect of the diffusion process which becomes unidirectional. So, the two-dimensional diffusion process can be treated in one-dimensional picture. On the other hand, we have computed the frequency-dependent conductivity σ( ω), of the two-dimensional system with Frenkel–Kontorova pair interaction in the intermediate friction regime, by using the continued fraction expansion method (CFEM). For the determination of the remainder of the development of the continued fraction we need to know an analytical expression of the DC conductivity σ(0), by other means. We are then led to derive an expression of σ(0), in the intermediate friction regime. Moreover, this new expression allows us to recover the two known results computed in the low and high friction limits. Its behaviour is compared with that obtained by Langevin dynamics. We have found a very good agreement between the two results. By varying the density of mobile ions we discuss commensurability effects. For commensurate cases, the resonance frequency of the conductivity is determined only by the external potential.

Stabilization of localized states in dissipative tunneling systems interacting with monochromatic fields

We study the dynamics of an initially localized symmetric two-level system coupled to high-temperature dissipative environments and driven by a strong time-periodic force which corresponds to high-frequency monochromatic light. Qualitative arguments based on the quantized representation of the radiation field predict a wealth of intriguing behaviors which are confirmed and quantified via accurate numerical path integral calculations. With the exception of very strong friction we find that high-frequency driving always helps stabilize localized states. At intermediate friction the delocalization rate approaches a “universal’’ limiting value which is largely independent of the parameters of the environment and of the specifics of the driving force, depending only on its overall strength. This robust behavior implies that localized states can be stabilized in these systems without much finetuning of external conditions. In the weak friction regime the interplay between phase interference and dissipation results in nonmonotonic variation of the decay rate with friction and driving frequency. The path integral results are compared to those obtained earlier via analytical treatments.

A study of the wear mechanism of diamond-like carbon films

In the present work, the tribological mechanism of diamond-like carbon (DLC) films was investigated. DLC films were coated on metallic substrates (M50 steel, Ti-6Al-4V alloy and AISI 440C steel) by methane ion-beam deposition. Pin-on-disc experiments showed that the DLC films possess excellent wear resistance and exhibit low values of friction coefficient ( ƒ < 0.1). The frictional response was characterized by an initial break-in period that was followed by an intermediate constant ƒ stage. The presence of a carbon transfer layer was observed on the wear scars of ball surfaces. Transmission electron microscopy and electron diffraction from the transfer layer showed that it contained a fine distribution of graphite nanoparticles (less than 5 nm) in a distorted diamond-like structure. Laser Raman spectra taken from the wear track region also revealed evidence of graphitization. In light of the present observations, the intermediate friction stage was mainly attributed to transfer layer formation. During long-duration wear experiments, a further reduction in ƒ was observed that finally reached steady state low friction ( ƒ = 0.05–0.07) and an ultralow wear rate (about 1.6 × 10 −9mm 3 m −1 N −1). Steady state was related to the complete transformation of diamond-like to graphite-like carbon under thermal and strain effects from the repeated friction through a precursor stage involving hydrogen evolution and sp 3 structure destabilization. The ultra-low wear rates were correlated to easy shear of low-strength atomic carbon interlayers in the graphitic structure.

Erratum: Theory of activated rate processes in the weak and intermediate friction cases: New analytical results for one and many degrees of freedom [J. Chem. Phys. 100, 7331 (1994)

First Page

Theory of activated rate processes in the weak and intermediate friction cases: New analytical results for one and many degrees of freedom

Simple analytical expressions for the reaction rate of activated rate processes are derived in the weak/intermediate friction limit for one and many degrees of freedom and for finite microcanonical reaction rates. The expressions are obtained by analytical solution of the steady-state integral master equations (in energy variables). The microcanonical reaction rate is taken to be independent of energy (higher than the activation energy). Irreversible transitions from one state and reversible transitions between many states are discussed in detail. A simple interpolation formula for the reaction rate is derived which describes a turnover from the weak friction regime to a strong friction one. The formula takes into account an interplay between activation and reaction at energies close to the activation energy. When applied to unimolecular gas phase reactions this interpolation formula bridges between the weak and strong collision limits. The formulas obtained are generalized to multidimensional activated rate processes.

Switch Phenomena in Molecular Three-Centre Electron Transfer Systems with Strong Electronic-Vibrational Coupling

Abstract Three-centre molecular and supramolecular electron transfer (ET) is important in device-like function of organized assemblies of bridge-group separated donor and acceptor units in solid-state or electrochemical systems. Other areas are solid-state phonon-kick charge capture, electrochemical ET via adsorbates or film impurities, large adsorbate STM, and long-range ET in redox metalloproteins. Optical three-level analogues are molecular Raman scattering and two-photon absorption. Three-level ET behaviour rests on coupling to environmental nuclear motion. A frame is second order perturbation theory including full vibrational manifolds in all three states. We discuss such a formalism. Common superex-change is recovered for high intermediate state energies while vibrationally unrelaxed state population emerges for low energies. These limits are separated by finite resonances in characteristic “switch” patterns. Intermediate state vibrational friction is briefly addressed and we note some biological and synthetic systems to which the formalism is appropriate.

Thermal activation: Kramers' theory revisited

The Brownian motion model of classical noise-induced escape from a metastable potential well is reconsidered, in particular for the friction dependence of the decay rate. Kramers' Fokker-Planck equation is reformulated in terms of the particle's energy and the action variable near the peak of the barrier. The ensuing probability density ϱ( s, ϵ) - and, hence, the decay rate Γ - is uniquely determined (i) by means of a spectral analysis and (ii) upon specifying the energy distribution of incoming particles. If such particles are taken to be absent, the decay rate goes to zero in the low friction limit according to Kramers' original formula, while for increasing friction it approaches the transition state value. The significance of diffusively re-entering particles for obtaining the correct high friction Kramers-Smoluchov-ski result is discussed. A problem with the underlying density is pointed out. The nature of the intermediate friction - the so-called turnover - regime is critically examined and a comparison is made with related recent work by Büttiker, Harris and Landauer, Mel'nikov and Meshkov, and Grabert.

Approximate and numerically exact solutions of the Fokker-Planck equation with bistable potentials

The kinetic rate for a symmetric bistable potential is calculated from the Fokker-Planck operator on both position and momentum. Numerical results are obtained by applying the Lanczos algorithm to the matrix representation of the time evolution operator. Both the continued fraction representation of the spectral density and the first positive eigenvalue which determines the transition rate, are obtained from the computed tridiagonal matrix. The numerical results are compared with the available analytical approximations of the kinetic rate for intermediate potential barriers. In particular, the localized functions method leads to an approximation which accurately describes the approach to the intermediate friction regime from the diffusion limit. Comparison is made also with the available approximations covering all the friction range, and derived under the asymptotic condition of very large potential barriers. A satisfactory agreement is found between the Mel'nikov-Meshkov equation and the numerical results for moderately large potential barriers.

Diffusion in a bistable potential. General inverse friction expansion

The derivation of the characteristic times and of the density probability distribution for the motion of a Brownian particle in a bistable potential at intermediate friction was, until now, essentially limited to low orders in the inverse frictionγ −1. On the other hand, at least for temperatures low with respect to the barrier height, the Kramers time, which is the lowest nonzero eigenvalue in the bistable potential problem, is known exactly. This paper presents a systematic approach for the determination of the solution of the Fokker-Planck equation in an arbitrary potential in the overdamped regime. This calculation includes anharmonicity corrections up to orderγ −5. One feature of this paper is to show that the problem is equivalent to replacing the original potentialφ(x) by a free energy which, for a velocity distribution at equilibrium, simply is $$\widetilde\phi $$ =φ(x) −k BT ln[g(x)], where $$g(x) = \left\{ {{{m\gamma } \mathord{\left/ {\vphantom {{m\gamma } {[2\phi ''(x)]}}} \right. \kern-\nulldelimiterspace} {[2\phi ''(x)]}}} \right\}\left\{ {1 - [{{1 - 4\phi ''(x)} \mathord{\left/ {\vphantom {{1 - 4\phi ''(x)} {m\gamma ^2 }}} \right. \kern-\nulldelimiterspace} {m\gamma ^2 }}]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$ For out-of-equilibrium velocity distribution an effective potential is also explicitly given. In every case the function g(x) plays a crucial role. This approach is then applied to the exact determination, in the low-temperature limit, of all the characteristic times and of the probability distribution in bistable potentials. Moreover, from the knowledge of the characteristic times and probability density distribution, it would be easy to determine the general and exact Suzuki scaling law for the relaxation from the instability point at intermediate friction.

Picosecond dynamics of barrier crossing in solution: A study of the conformational change of excited state 1,1′-binaphthyl

The rates of optically induced conformational changes of excited state 1,1′-binaphthyl in a series of alcohol solvents were measured by a picosecond laser technique. Excellent agreement with experiment is found for the Kramers model for both the intermediate and high friction regimes using a frequency independent friction coefficient evaluated from classical hydrodynamics. This is the first observation of Kramers behavior in the intermediate friction regime and indicates that deviations from Kramers behavior is not a universal phenomenon in the conformational dynamics of flexible molecules in solution. It also emphasizes the possibility that the observed deviations from the Kramers model can be due to effects other than non-Markovian friction.

Brownian motion in the bistable potential at intermediate and high friction: Relaxation from the instability point

We consider a Brownian particle in a double-well potential at intermediate and high friction and study the evolution of an initial distribution which is located at the top of the barrier. The evolution equation is obtained from the Fokker-Planck-Klein-Kramers equation by an inverse friction expansion and is transformed to a time-dependent Schrödinger equation which we solve approximately using the WKB method. The result becomes exact for temperatures low compared with the barrier height. We calculate explicitly the characteristic time in which the distribution changes from a single peak to a double peak structure and show that the decay process is mainly described by a scaling-function which reduces to Suzuki's scaling solution in the limit of high friction.

Diffusion in a bistable potential at intermediate and high friction

We study the motion of a Brownian particle in a bistable potential for intermediate and high-friction γ. Following ideas of Titulaer we perform a high-friction expansion of the distribution functionP(v,x,t) in velocity and space. We show (for arbitrary potential) that the expansion coefficients obey simple recursion relations, which allow them to be calculated easily. When terms of order γ−5 are neglected the resulting differential equations can be transformed into Hermitian Schrodinger-type equations. Using the WKB technique we solve these equations analytically for the case of the bistable potential and discuss the various time regimes involved in the system, in particular we show that the final approach to equilibrium is governed by the Kramers rate. Our results become exact in the limit of low temperatures.

Photochemical isomerization in solution. Photophysics of diphenyl butadiene

We have examined the radiative and nonradiative decay rates of diphenyl butadiene in hydrocarbon and polar solvents as a function of temperature and pressure. We demonstrate that the radiative rate constant has a small dependence on solvent polarizability, in contrast to what is observed for diphenylhexatriene. We make an explicit separation of the internal molecular barrier to isomerization and the barrier due to frictional forces in the solvent. The energy of the internal barrier to isomerization is less than the activation energy observed in solution, and depends on the polarizability and polarity of the solvent. In contrast to diphenylhexatriene, it decreases, in more polar solvents. We compare the viscosity dependence of the barrier crossing process with Kramers’ theory, the recent kinetic theory result of Skinner and Wolynes [J. Chem. Phys. 69, 2143 (1978)], and the free volume theory of Gegiou, Muzkat, and Fischer, [J. Amer. Chem. Soc. 90, 12 (1968)]. Our data clearly show that the isomerization process is in the intermediate friction limit and we obtain experimental values for the parameters of the three theories. Over the viscosity region studied, Kramers’ and Skinner and Wolynes’ equations coincide exactly and do not reproduce the functional dependence of the radiationless rate pre-exponential factor as well as the free volume theory. We speculate that it may be necessary to consider the frequency dependence of medium viscosity.

Eigenvalues and eigenfunctions of the Kramers equation. Application to the Brownian motion of a pendulum

The Brownian motion of s particle in a potential is described by a Fokker-Planck equation with two variables (position and velocity), i.e. by the Kramers equation. Eigenvalues and eigenfunctions can be obtained by a matrix continued fraction method. This method is applied to the Brownian motion of a pendulum, where the force is proportional to sin x. The real and complex eigenvalues with small real parts are plotted as a function of the damping constant. In the low friction limit the real parts of the complex eigenvalues are proportional to the square of the friction constant, whereas the real eigenvalues are proportional to the friction constant itself. In the high friction limit the eigenvalues are real and the lowest ones are proportional to the inverse friction constant. In the intermediate friction regime a complicated mixture of real and complex eigenvalues is found. The same method is also applied to the Boltzmann equation with the BGK collision operator.

Bistability effects of the brownian motion in periodic potentials

In the stationary state the equation of motion for particles moving in a periodic potential has two solutions, a locked one and a running one, for low and intermediate damping constants and for suitable external forces. The effect of an additional Langevin force to this bistable behaviour is investigated. For finite noise strength, the mobility depends continuously on the external force, whereas in the limit of vanishing strength of the noise force one gets a sharp transition between the locked and the running solution at a critical external force. This critical force is calculated exactly in the low friction limit and approximately for intermediate friction constants. Furthermore the temperature dependence for various forces including the critical one is shown in the low friction limit.

Internal friction of compressed pharmaceutical powders observed in terms of the die wall pressure.

The internal friction of powders was examined during tablet preparation for lactose and phenacetin as base materials and for potassium chloride, potassium bromide and sodium chloride as model samples. The coefficient of internal friction decreased with the porosity, even if the voids were filled by fragmentation of particles or by plastic flow. When the porosity had reached zero, the state of the whole bed of powder was found to be plastic, where the coefficient of internal friction was zero, for magnesium stearate, phenacetin, potassium bromide and potassium chloride. The plastic behavior of magnesium stearate and phenacetin was considered to be related to the capping of tablets containing them. For lactose granules, the porosity did not reach zero even at a compacting pressure of 4000 kg/cm2, the maximum used in this work, because of its crystal hardness. The coefficient of internal friction was large and decreased only gradually with the porosity. Mixtures of two components exhibited intermediate internal friction, corresponding to the composition. However, when hydroxypropyl cellulose (HPC) was added to phenacetin powder as a binder, HPC increased the yield stress from 2000 to 3100 kg/cm2 even at a content of only 2% w/w on a dry basis. In contrast, lubricants (magnesium stearate and talc) added to granules had little or no effect on the internal friction, although this small amount was sufficient to reduce the wall friction.

Two particle model for the diffusion of interacting particles in periodic potentials

The random motion of two interacting particles in a periodic potential with a finite number of sites is investigated as a model that may be applied to superionic conductivity. Starting from the Fokker-Planck-equation for the model and using an appropriate series expansion for the probability density, solutions for the frequency dependent conductivity are given. Explicit numerical results are shown for hard-core and Coulomb interaction in the range of intermediate and high friction constants.

Transition Behavior of a Blasius-Type Boundary Layer Subjected to Uniform Blowing or Suction

The momentum integral equation is used to study the transient behavior of a Blasius-type boundary layer which is suddenly subjected to uniform blowing or suction. The time required for the boundary layer to adjust itself from one steady state (Blasius) to the other (constant blowing or suction) was found to be proportional to the distance from the leading edge. Boundary-layer thickness of intermediate states and skin friction coefficients are also reported.